Thermalization of Local Observables in the $$\alpha $$-FPUT Chain

Ganapa, Santhosh; Apte, Amit; Dhar, Abhishek

doi:10.1007/s10955-020-02576-2

Cited by 12 publications

(12 citation statements)

References 104 publications

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“…Thus, α is the origin of nonlinearity in the system. Note that the cubic potential of the α-FPUT system implies that the system stays bounded only if the total energy is sufficiently small and the precise condition is E < µ 3 /(6α 2 ), corresponding to all energy contributing to the potential energy of a single particle [13]. In practice this is highly improbable and one can work with energies slightly higher than this bound.…”

Section: A the Fput Problemmentioning

confidence: 99%

“…As far as the question of if and when the FPUT chain thermalizes is concerned, there have been many studies done to investigate this. Some of the methods include the closeness to integrability and soliton solutions [3], stochasticity threshold [4,5], Lyapunov exponent studies [6][7][8], breather solutions [9][10][11][12] and local equilibration studies [13]. More recently the formalism of wave turbulence [14,15] was applied to the FPUT chain [16][17][18][19] in order to explain the dependence of equilibration time on the nonlinearity parameter.…”

Section: Introductionmentioning

confidence: 99%

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Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Ganapa¹

2023

Preprint

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The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α−FPUT system in more detail. We aim to reconstruct the quasiperiodic behaviour observed in the original paper from the canonical transformation used to remove the three wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work we show that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α−FPUT system, and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α−FPUT problem. We argue that certain aspects of the α−FPUT system such as presence of the stochasticity threshold, thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.

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Section: A the Fput Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Ganapa¹

2023

Preprint

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“…While in non-pathological cases these two kind of averages are usually expected to yield the same kind of results, there are situations where the choice of initial conditions and the choice of the averaging procedure are of crucial importance, in particular for the behaviour of transients. This issue has been investigated in depth for the FPUT model [21][22][23][24], for which it has been shown explicitly how some particular choices of initial conditions, e.g., the choice of Fourier modes initial phases [22], strongly affect the duration of transients. Quite remarkably, in [24] it was then presented a first evidence that an appropriate choice of conditions (or, more generally, an appropriate choice of degrees of freedom as we will show in the following) allows to find thermalization even in the Toda model, a result in perfect agreement with the recent findings of two of us [9].…”

Section: Random Modes Thermalizationmentioning

confidence: 99%

“…This issue has been investigated in depth for the FPUT model [21][22][23][24], for which it has been shown explicitly how some particular choices of initial conditions, e.g., the choice of Fourier modes initial phases [22], strongly affect the duration of transients. Quite remarkably, in [24] it was then presented a first evidence that an appropriate choice of conditions (or, more generally, an appropriate choice of degrees of freedom as we will show in the following) allows to find thermalization even in the Toda model, a result in perfect agreement with the recent findings of two of us [9]. Coming back to the definition of n eff (θ , t), which, physically, is analogous to an inverse participation ratio, we have that it reads as:…”

Section: Random Modes Thermalizationmentioning

confidence: 99%

Thermalization without chaos in harmonic systems

Cocciaglia

Vulpiani

Gradenigo

2022

Physica A: Statistical Mechanics and its Applications

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“…At the same time, the equipartition of observables weakly correlated to the actions of an integrable limit may show fast equilibration, as studied e.g. for local energies of an FPUT model [17]. Thus, the choice of observables impacts the thermalization timescale analysis.…”

Section: Introductionmentioning

confidence: 99%

Thermalization dynamics of macroscopic weakly nonintegrable maps

Malishava,

Flach

2022

Preprint

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We study thermalization of weakly nonintegrable nonlinear unitary lattice dynamics. We identify two distinct thermalization regimes close to the integrable limits of either linear dynamics or disconnected lattice dynamics. For weak nonlinearity the almost conserved actions correspond to extended observables which are coupled into a long-range network. For weakly connected lattices the corresponding local observables are coupled into a short-range network. We compute the evolution of the variance σ 2 (T ) of finite time average distributions for extended and local observables. We extract the ergodization time scale TE which marks the onset of thermalization, and determine the type of network through the subsequent decay of σ 2 (T ). We use the complementary analysis of Lyapunov spectra (arXiv:2109.01361) and compare the Lyapunov time TΛ with TE. We characterize the spatial properties of the tangent vector and arrive at a complete classification picture of weakly nonintegrable macroscopic thermalization dynamics.

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Thermalization of Local Observables in the $$\alpha $$-FPUT Chain

Cited by 12 publications

References 104 publications

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Quasiperiodicity in the $α-$Fermi-Pasta-Ulam-Tsingou system revisited: an approach using ideas from wave turbulence

Thermalization without chaos in harmonic systems

Thermalization dynamics of macroscopic weakly nonintegrable maps

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